Business Forecasting;Using the United Kingdom statistics locates Banks consumer credit: Gross lending figures from 1993-2003

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1. Examine the data by plotting and/or otherwise for seasonal effects, trends and cycles.

Year Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

1 454 61 454 4405 4788 4576 41 45 478714 41 80 487 46 4657 457 464 551 48 4688 505 57

15 481 4571 5485 508 56 5678 5587 6116 556 576 5781 5855

16 576 5448 577 650 658 60 7066 6841 6480 617 6468 7161

17 6 6088 646 78 700 747 7 7516 750 74 71 8751

18 7516 77 86 8180 87 884 17 8606 4 878 80 445

1 8107 84 767 88 8511 874 85 80 87 460 1057 10450

000 50 1006 108 4 11156 10806 1077 1088 06 10857 108 1070

001 1114 100 1106 117 1188 1171 1 15 1158 147 184 165

00 176 1165 167 184 14085 111 147 115 18 144 14 1

00 10 110

Resource http//www.statistics.gov.uk/statbase/TSDtimezone.asp (Accessed /04/0)

The figure shows Banks consumer credit Gross lending on a monthly basis starting with the first month of 14 and ending with the twelfth month of 00. And we will use the data in 00 to check on the forecast.

As seen from the graph above, the time series data moves upward over a period of time so it can be said that the data shows a fairly strongly positive trend, so that we might expect high autocorrelation coefficients. Considering ACF, it supports this expectation because it is significantly different from zero. Autocorrelation at lag 1 is 0.45 and move downwards to 0.581 at lag 16.

Afterwards, we will choose data from 14-001 to model.

. Try a classical decomposition method on part of the data and check it against the rest of the data.

The ACF of the actual data shows that there is a trend or cycle or both, so we should look at the ACF of the first difference.

From the ACF of the first difference, we could use the 1 points moving average to smooth the data. Subsequently, we will calculate the twelve point moving average, seasonal estimate, trend-cycle estimate and forecast based on the trend-cycle and the seasonal ratios.

After calculating center moving average and getting seasonal estimate (ratio), we will find out the factors as following table

Sequence Mean Count

1 0.5 7

0.1 7

1.00 7

4 0.8 7

5 1.01 7

6 1.0 7

7 1.04 7

8 1.05 7

1.00 7

10 1.00 7

11 1.00 7

1 1.04 7

Then we can de-seasonalise the data by dividing the factors into the data to create a trend cycle column. Next, we regress the trend cycle against the time and will get a trend column.

Coefficients

Unstandardized Coefficients Standardized Coefficients t Sig.

Model B Std. Error Beta

1 (Constant) 45.164 84.847 46.48 0.000

TIME 84.57 1.51 0.85 55.56 0.000

a Dependent Variable TRENCYC

From the coefficients table above, the trend equation is

Trend = 45.164+84.57 Time

Then we can create a moving average forecasting column with the factors and trend column. The moving average forecasting equation is

Forecast = Trend Factors

Therefore, it can be implied that

Forecast = (45.164+84.57 Time) Factors

Moreover, the errors will be calculated by deducting the forecast data from the actual data in order to see how accurate the forecast is.

Errors = Data Forecast

See the figures of the result as following table

Time Data Center Moving Average Ratios Sequence Factors Trend-cycle Trend line Forecast Errors

14 Jan 1 41 0.5 446. 40.5 88.04 00.6

Feb 80 0.1 47.6 411.88 74.6 6.7

Mar 487 1.00 487.00 418.4 418.4 680.76

Apr 4 46 0.8 4485.71 48.5 416.4 1.06

May 5 4657 1.01 4610.8 466.5 4410.6 46.8

Jun 6 457 1.0 485.80 4451.1 4540. 416.67

Jul 7 464 4804.00 0.8 7 1.04 451.46 455.66 4717.0 -.0

Aug 8 551 4857.1 1.14 8 1.05 550.48 455.66 4851.0 661.8

Sep 48 407.00 1.0 1.00 550.48 4704.8 4704.8 84.6

Oct 10 4688 461.50 0.4 10 1.00 4688.00 4788.7 4788.7 -100.7

Nov 11 505 50.88 1.00 11 1.00 505.00 487.0 487.0 178.1

Dec 1 57 5107.04 1.05 1 1.04 5165.8 457.45 5155.75 16.5

15 Jan 1 481 5107.04 0. 1 0.5 5066. 5041.81 478.71 .

Feb 14 4571 56.6 0.87 1 0.1 50.08 516.16 4664.81 -.81

Mar 15 5485 585.6 1.04 1.00 5485.00 510.5 510.5 74.48

Apr 16 508 554. 0.5 4 0.8 50.04 54.88 5188.8 -0.8

May 17 56 54.46 1.05 5 1.01 50.04 57. 54.0 58.7

Jun 18 5678 547.6 1.04 6 1.0 5566.67 546.5 557.86 105.14

Jul 1 5587 55.6 1.01 7 1.04 57.1 5547.5 576.86 -18.86

Aug 0 6116 5615.71 1.0 8 1.05 584.76 56.0 51. 0.08

Sep 1 556 5664.5 0.8 1.00 556.00 5716.66 5716.66 -154.66

Oct 576 578.4 1.01 10 1.00 576.00 5801.0 5801.0 -8.0

Nov 5781 5817.67 0. 11 1.00 5781.00 5885.8 5885.8 -104.8

Dec 4 5855 586.54 1.00 1 1.04 56.81 56.7 608.5 -5.5

16 Jan 5 576 545.6 0.7 1 0.5 6065.6 6054.0 5751.8 10.6

Feb 6 5448 607.7 0.0 0.1 586.81 618.45 5751.8 -17.

Mar 7 577 6106.5 0.5 1.00 577.00 6.80 6.80 -44.80

Apr 8 650 61.58 1.0 4 0.8 647.5 607.16 6181.0 168.8

May 658 66. 1.05 5 1.01 6516.8 61.5 6455.4 16.57

Jun 0 60 65. 0.5 6 1.0 514.71 6475.87 6605. -57.

Jul 1 7066 64.00 1.10 7 1.04 674. 6560. 68.64 4.6

Aug 6841 6485. 1.05 8 1.05 6515.4 6644.5 676.8 -15.8

Sep 6480 65. 0. 1.00 6480.00 678.5 678.5 -48.4

Oct 4 617 655.00 1.05 10 1.00 617.00 681.0 681.0 10.70

Nov 5 6468 6655.7 0.7 11 1.00 6468.00 687.66 687.66 -4.66

Dec 6 7161 67.58 1.06 1 1.04 6885.58 68.0 761.0 -100.0

17 Jan 7 6 68. 0.4 1 0.5 678.4 7066.7 671.05 -1.05

Feb 8 6088 68.1 0.88 0.1 660.11 7150.7 6507.16 -41.16

Mar 646 664.58 0.0 1.00 646.00 75.0 75.0 -8.0

Apr 40 78 7050.67 1.05 4 0.8 75.65 71.44 717.06 08.4

May 41 700 711.88 0.8 5 1.01 6.60 740.80 7477.84 -468.84

Jun 4 747 76.58 1.0 6 1.0 76.47 7488.16 767. -164.

Jul 4 7 74.67 1.08 7 1.04 7618.7 757.5 7875.4 47.58

Aug 44 7516 7448.54 1.01 8 1.05 7158.10 7656.87 80.7 -5.7

Sep 45 750 7588.7 0. 1.00 750.00 7741. 7741. -1.

Oct 46 74 7710.5 1.0 10 1.00 74.00 785.5 785.5 117.41

Nov 47 71 774.5 0.5 11 1.00 71.00 70.4 70.4 -518.4

Dec 48 8751 707.6 1.11 1 1.04 8414.4 74.0 814.07 46.

18 Jan 4 7516 801.08 0.4 1 0.5 711.58 8078.66 7674.7 -158.7

Feb 50 77 8116.67 0.0 0.1 806.64 816.01 748.4 -1.4

Mar 51 86 8.8 1.0 1.00 86.00 847.7 847.7 115.6

Apr 5 8180 88. 0.8 4 0.8 846.4 81.7 8165.0 14.1

May 5 87 844.71 0.8 5 1.01 8145.54 8416.0 8500.5 -7.5

Jun 54 884 857. 1.05 6 1.0 8807.84 8500.44 8670.45 1.55

Jul 55 17 8580.88 1.06 7 1.04 8775.6 8584.80 88.1 18.81

Aug 56 8606 864.46 1.00 8 1.05 816.1 866.16 10.61 -46.61

Sep 57 4 877. 1.06 1.00 4.00 875.51 875.51 488.4

Oct 58 878 88.6 0. 10 1.00 878.00 887.87 887.87 -10.87

Nov 5 80 886.67 1.01 11 1.00 80.00 8. 8. -.

Dec 60 445 811.58 1.06 1 1.04 081.7 006.58 66.85 78.15

1 Jan 61 8107 878.88 0.0 1 0.5 85.68 00.4 866. -5.

Feb 6 84 060.08 0.1 0.1 07.6 175.0 84.5 -15.5

Mar 6 767 14.1 1.07 1.00 767.00 5.66 5.66 507.4

Apr 64 88 0.67 0.6 4 0.8 01.7 44.01 157.1 -4.1

May 65 8511 04.04 0.1 5 1.01 846.7 48.7 5.65 -1011.65

Jun 66 874 415.7 1.05 6 1.0 680. 51.7 70.8 171.0

Jul 67 85 516.6 1.04 7 1.04 47.08 57.08 80.7 -18.7

Aug 68 80 651. 1.0 8 1.05 61.0 681.44 10165.51 -5.51

Sep 6 87 770.8 1.0 1.00 87.00 765.80 765.80 1.0

Oct 70 460 86.50 0.6 10 1.00 460.00 850.15 850.15 -0.15

Nov 71 1057 67.6 1.06 11 1.00 1057.00 4.51 4.51 66.4

Dec 7 10450 10117.00 1.0 1 1.04 10048.08 10018.87 1041.6 0.8

000 Jan 7 50 1014.1 0. 1 0.5 1001.58 1010. 58.06 -68.06

Feb 74 1006 1076.4 0.8 0.1 11017.58 10187.58 70.70 755.0

Mar 75 108 1016.88 1.05 1.00 108.00 1071.4 1071.4 561.06

Apr 76 4 1071.71 0.0 4 0.8 5.67 1056.0 1014.17 -806.17

May 77 11156 1044.1 1.07 5 1.01 11045.54 10440.65 10545.06 610.4

Jun 78 10806 10465. 1.0 6 1.0 1054.1 1055.01 1075.51 70.4

Jul 7 1077 1054. 1.0 7 1.04 1058.65 1060.7 110.74 -60.74

Aug 80 1088 10610.8 1.0 8 1.05 106.81 106.7 118.41 -46.41

Sep 81 06 1061.7 0. 1.00 06.00 10778.08 10778.08 -87.08

Oct 8 10857 10707.8 1.01 10 1.00 10857.00 1086.44 1086.44 -5.44

Nov 8 108 10817.5 1.01 11 1.00 108.00 1046.80 1046.80 -54.80

Dec 84 1070 1088.1 0.8 1 1.04 107.1 1101.15 1147.40 -76.40

001 Jan 85 1114 1100.08 1.01 1 0.5 117.47 11115.51 1055.7 58.7

Feb 86 100 11170.04 0.0 0.1 11014. 111.87 1011.88 -168.88

Mar 87 1106 110.67 0.8 1.00 1106.00 1184. 1184. -.

Apr 88 117 11450.75 0.8 4 0.8 11456.1 1168.58 11141.1 85.7

May 8 1188 1161.88 1.0 5 1.01 11780.0 1145.4 11567.47 0.5

Jun 0 1171 117.67 1.00 6 1.0 1155.80 1157. 11768.04 .6

Jul 1 1 1.04 141.7 1161.65 1086.5 84.48

Aug 15 1.05 1178.10 11706.01 11.1 .6

Sep 1158 1.00 1158.00 1170.7 1170.7 -4.6

Oct 4 147 1.00 147.00 11874.7 11874.7 17.8

Nov 5 184 1.00 184.00 115.08 115.08 88.

Dec 6 165 1.04 114.04 104.44 155.17 10.8

Then, we can plot the decomposition forecasting against the actual data to illustrate what is happening.

The graph shows that the decomposition forecasting fits the actual data better at the middle. However, at the beginning and the end, there are some errors.

Comparing the forecast data with the rest of the data, we calculate them by extending the time line and see the result as following table

Rest Data Time Factors Trend Forecast Errors

00 Jan 176.00 7 0.5 117.7 1151.40 140.60

Feb 1165.00 8 0.1 11.15 1111.06 851.4

Mar 167.00 1.00 16.51 16.51 8.4

Apr 184.00 100 0.8 180.86 11.5 1850.75

May 14085.00 101 1.01 1465. 158.87 145.1

Jun 111.00 10 1.0 154.58 1800.57 -688.57

Jul 147.00 10 1.04 16.4 11. 158.71

Aug 115.00 104 1.05 1718. 154.1 -1.1

Sep 18.00 105 1.00 180.65 180.65 186.5

Oct 144.00 106 1.00 1887.01 1887.01 606.

Nov 14.00 107 1.00 171.6 171.6 -.6

Dec 1.00 108 1.04 1055.7 1577.5 44.05

00 Jan 158.00 10 0.5 1140.08 148.07 1114.

Feb 17.00 110 0.1 14.4 104. 7.77

Sum of squared errors = 1855.51

Mean-squared error (MSE) = 1108.75

Root-mean-squared error (RMSE) = 57.74

This RMSE is about 7. % of the mean for the rest of data during the forecast period

From the table above, we can notice the errors as illustrated the graph below which is plotted from rest data against forecast data of 00-00.

Although there is some significant errors, perhaps which are caused by economic uncertainty during that period, overall the plot graph below shows that the predict data relatively fit the real data.

. Try an Autocorrelation approach on the same data and discuss the differences between the results and those from the Decomposition approach. Please comment on the different from of the equation for the forecast.

After the first difference, it shows that there are seven spikes on lag1, lag, lag4, lag10, lag1, lag1, and lag14 so we will try on these lagged data to calculate correlations.

The table below shows the correlations between the actual data and lagged data

Correlations

DATA LAGS LAGS LAGS LAGS LAGS LAGS LAGS

(DATA,1) (DATA,) (DATA,4) (DATA,10) (DATA,1) (DATA,1) (DATA,14)

DATA Pearson Correlation 1 .5 .6 .5 .6 .7 .40 .54

Sig. (-tailed) . .000 .000 .000 .000 .000 .000 .000

N 6 5 86 84 8 8

LAGS(DATA,1) Pearson Correlation .5 1 .56 .6 .50 5 .7 .8

Sig. (-tailed) .000 . .000 .000 .000 .000 .000 .000

N 5 5 86 84 8 8

LAGS(DATA,) Pearson Correlation .6 .56 1 .57 .8 .5 .4 .58

Sig. (-tailed) .000 .000 . .000 .000 .000 .000 .000

N 86 84 8 8

LAGS(DATA,4) Pearson Correlation .5 .6 .57 1 .45 .51 .51 .

Sig. (-tailed) .000 .000 .000 . .000 .000 .000 .000

N 86 84 8 8

LAGS(DATA,10) Pearson Correlation .6 .50 .8 .45 1 .48 .64 .

Sig. (-tailed) .000 .000 .000 .000 . .000 .000 .000

N 86 86 86 86 86 84 8 8

LAGS(DATA,1) Pearson Correlation .7 .5 .5 .51 .48 1 .48 .45

Sig. (-tailed) 0.000 .000 .000 .000 .000 . .000 .000

N 84 84 84 84 84 84 8 8

LAGS(DATA,1) Pearson Correlation .40 .7 .4 .51 .64 .48 1 .47

Sig. (-tailed) .000 .000 .000 .000 .000 .000 . .000

N 8 8 8 8 8 8 8 8

LAGS(DATA,14) Pearson Correlation .54 .8 .58 . . .45 .47 1

Sig. (-tailed) .000 .000 .000 .000 .000 .000 .000 .

N 8 8 8 8 8 8 8 8

Correlation is significant at the 0.01 level (-tailed).

From the correlation between the data and lagged data, these are all high and then we need to use regression on the chosen columns to find the relationship.

Variables Entered/Removed

Model Variables Entered Variables Removed Method

1 LAGS(DATA,14), LAGS(DATA,4), LAGS(DATA,10), LAGS(DATA,1), LAGS(DATA,), LAGS(DATA,1), LAGS(DATA,1) . Enter

a All requested variables entered.

b Dependent Variable DATA

Model Summary

Model R R Square Adjusted R Square Std. Error of the Estimate

1 0.80 0.61 0.57 448.111

a Predictors (Constant), LAGS(DATA,14), LAGS(DATA,4), LAGS(DATA,10), LAGS(DATA,1), LAGS(DATA,), LAGS(DATA,1), LAGS(DATA,1)

This explains 5.7% of the variability and so will be reasonable fit to the data

Coefficients

Unstandardized Coefficients Standardized Coefficients t Sig.

Model B Std. Error Beta

1 (Constant) 1.580 10.57 1.85 .067

LAGS(DATA,1) 1.658E-0 .1 .017 .16 .8

LAGS(DATA,) .10 .10 .01 .04 .00

LAGS(DATA,4) 1.70E-0 .0 .00 .017 .86

LAGS(DATA,10) -.11 .10 -.11 -1.17 .4

LAGS(DATA,1) .5 .105 .547 5.66 .000

LAGS(DATA,1) 6.E-0 .14 .05 .471 .6

LAGS(DATA,14) .16 .104 .181 1.88 .064

a Dependent Variable DATA

From the coefficients table, the lag1, lag4, lag10, lag1, and lag14 data are not significant as well as the constant so the regression should be repeated only with lag and lag1 data because their significant is less than .05

Variables Entered/Removed

Model Variables Entered Variables Removed Method

1 LAGS(DATA,1), LAGS(DATA,) . Enter

a All requested variables entered.

b Dependent Variable DATA

c Linear Regression through the Origin

Model Summary

Model R R Square Adjusted R Square Std. Error of the Estimate

1 . .7 .7 44.446

a For regression through the origin (the no-intercept model), R Square measures the proportion of the variability in the dependent variable about the origin explained by regression. This CANNOT be compared to R Square for models which include an intercept.

b Predictors LAGS(DATA,1), LAGS(DATA,)

This fit has increased slightly from .7% to 6% by removing the lag1, lag4, lag10, lag1, and lag14 data as well as the constant.

Coefficients

Unstandardized Coefficients Standardized Coefficients

Model B Std. Error Beta t Sig.

1 LAGS(DATA,) .4 .07 .44 6.08 .000

LAGS(DATA,1) .651 .07 .575 8.70 .000

a Dependent Variable DATA

b Linear Regression through the Origin

All lag data are significant, so

Forecast = 0.4 (data lagged) + 0.651 (data lagged1)

See the figures of the Autocorrelation forecasting as following table

Time Data lag lag1 Forecast Error

14 Jan 1 41

Feb 80

Mar 487

Apr 4 46 41.00

May 5 4657 80.00

Jun 6 457 487.00

Jul 7 464 46.00

Aug 8 551 4657.00

Sep 48 457.00

Oct 10 4688 464.00

Nov 11 505 551.00

Dec 1 57 48.00

15 Jan 1 481 4688.00 41.00 4746.01 66.

Feb 14 4571 505.00 80.00 4808.81 -7.81

Mar 15 5485 57.00 487.00 554.54 -4.54

Apr 16 508 481.00 46.00 474.70 1.0

May 17 56 4571.00 4657.00 508.8 65.6

Jun 18 5678 5485.00 457.00 564. 4.08

Jul 1 5587 508.00 464.00 5.8 .18

Aug 0 6116 56.00 551.00 6087.75 8.5

Sep 1 556 5678.00 48.00 5740.48 -178.48

Oct 576 5587.00 4688.00 5504.58 58.4

Nov 5781 6116.00 505.00 57.78 -1.78

Dec 4 5855 556.00 57.00 58.8 -8.8

16 Jan 5 576 576.00 481.00 566. 8.78

Feb 6 5448 5781.00 4571.00 551.58 -65.58

Mar 7 577 5855.00 5485.00 6141.08 -68.08

Apr 8 650 576.00 508.00 5848. 501.68

May 658 5448.00 56.00 607.16 484.84

Jun 0 60 577.00 5678.00 60.7 -17.7

Jul 1 7066 650.00 5587.00 644.7 641.1

Aug 6841 658.00 6116.00 6871.01 -0.01

Sep 6480 60.00 556.00 66.5 10.65

Oct 4 617 7066.00 576.00 685.6 6.1

Nov 5 6468 6841.00 5781.00 6766.6 -8.6

Dec 6 7161 6480.00 5855.00 6656. 504.68

17 Jan 7 6 617.00 576.00 6787.6 -5.6

Feb 8 6088 6468.00 5448.00 686.10 -8.10

Mar 646 7161.00 577.00 601.0 -655.0

Apr 40 78 6.00 650.00 6.4 44.06

May 41 700 6088.00 658.00 657.51 51.4

Jun 4 747 646.00 60.00 666.48 80.5

Jul 4 7 78.00 7066.00 7840.66 8.4

Aug 44 7516 700.00 6841.00 750.44 -14.44

Sep 45 750 747.00 6480.00 74.1 0.87

Oct 46 74 7.00 617.00 781.16 -8.16

Nov 47 71 7516.00 6468.00 7510.1 -11.1

Dec 48 8751 750.00 7161.00 76.0 787.1

18 Jan 4 7516 74.00 6.00 7648.17 -1.17

Feb 50 77 71.00 6088.00 707.4 1.06

Mar 51 86 8751.00 646.00 707.84 455.17

Apr 5 8180 7516.00 78.00 8105.1 74.7

May 5 87 77.00 700.00 778.80 44.0

Jun 54 884 86.00 747.00 856.8 447.7

Jul 55 17 8180.00 7.00 8748.8 78.11

Aug 56 8606 87.00 7516.00 8504.57 101.4

Sep 57 4 884.00 750.00 88.50 40.50

Oct 58 878 17.00 74.00 177.65 -44.65

Nov 5 80 8606.00 71.00 858.58 0.4

Dec 60 445 4.00 8751.00 754.14 -0.14

1 Jan 61 8107 878.00 7516.00 874.51 -617.51

Feb 6 84 80.00 77.00 86.7 -468.7

Mar 6 767 445.00 86.00 50.67 176.

Apr 64 88 8107.00 8180.00 8884.15 -51.15

May 65 8511 84.00 87.00 866.11 -455.11

Jun 66 874 767.00 884.00 1016.0 -6.0

Jul 67 85 88.00 17.00 81.6 .64

Aug 68 80 8511.00 8606.00 8.84 41.16

Sep 6 87 874.00 4.00 1051. -64.

Oct 70 460 85.00 878.00 10006.6 -546.6

Nov 71 1057 80.00 80.00 101. 474.71

Dec 7 10450 87.00 445.00 105. -8.

000 Jan 7 50 460.00 8107.00 40.60 .40

Feb 74 1006 1057.00 84.00 10005.1 0.0

Mar 75 108 10450.00 767.00 1045.87 -11.87

Apr 76 4 50.00 88.00 .5 -50.5

May 77 11156 1006.00 8511.00 4.08 11.

Jun 78 10806 108.00 874.00 1118.66 -77.66

Jul 7 1077 4.00 85.00 10515. 57.77

Aug 80 1088 11156.00 80.00 116.81 -414.81

Sep 81 06 10806.00 87.00 1145.7 -1.7

Oct 8 10857 1077.00 460.00 10887.81 -0.81

Nov 8 108 1088.00 1057.00 11675.85 -78.85

Dec 84 1070 06.00 10450.00 11151.68 -44.68

001 Jan 85 1114 10857.00 50.00 1070.5 17.75

Feb 86 100 108.00 1006.00 1108.51 -185.51

Mar 87 1106 1070.00 108.00 1175.5 -61.5

Apr 88 117 1114.00 4.00 1074.07 5.

May 8 1188 100.00 11156.00 1166.65 5.5

Jun 0 1171 1106.00 10806.00 1180. -.

Jul 1 1 117 1077 1141.88 87.1

Aug 15 1188 1088 107.40 17.60

Sep 1158 1171 06 1165.06 -67.06

Oct 4 147 1 10857 174.74 50.6

Nov 5 184 15 108 1501.7 47.6

Dec 6 165 1158 1070 1157.7 677.8

Then, we can plot the autocorrelation forecasting against the actual data to illustrate what is happening.

The graph shows that the Autocorrelation forecasting fits the actual data better at the middle. However, at the end, there are some errors.

Comparing the forecast data with the rest of the data, we calculate them by extending the time line and see the result as following table

Time Data lag lag1 Forecast Error

00 Jan 7 176 147 1114 106.5 -07.5

Feb 8 1165 184 100 1165.68 -00.68

Mar 167 165 1106 1748.1 -6.1

Apr 100 184 176 117 111. 107.71

May 101 14085 1165 1188 18. 1086.77

Jun 10 111 167 1171 14.0 -110.0

Jul 10 147 184 1 14555.76 167.4

Aug 104 115 14085 15 1406.8 -1.8

Sep 105 18 111 1158 1711. 77.77

Oct 106 144 147 147 15087.1 -15.1

Nov 107 14 115 184 14166.08 -14.08

Dec 108 1 18 165 17.56 -5.56

00 Jan 10 158 144 176 141. -6.

Feb 110 17 14 1165 1470.75 -118.75

Sum of squared errors = 1071844.

Mean-squared error (MSE) = 7650.16

Root-mean-squared error (RMSE) = 874.76

This RMSE is about 6.6 % of the mean for the rest of data during the forecast period

From the table above, we can notice the errors as illustrated the graph below which is plotted from rest data against forecast data of 00-00.

However, there are some errors at the end, overall the plot graph below (During 14-00) shows that the predict data relatively fit the real data.

From the graph above, it illustrated the actual data (red line), Decomposition forecasting (blue line), and Autocorrelation forecasting (green line). Some periods Decomposition model is closer and fitter than Autocorrelation model, but some periods do contrast. Therefore, it is relatively difficult to say which model is better because both models are quite close to the real data.

However, it can be seen the error of these two models as following graph

Decomposition Model

Forecast = (45.164 + 84.57 Time) Factors

Autocorrelation Model

Forecast = 0.4 (data lagged) + 0.651 (data lagged1)

From these equations, it demonstrates that decomposition model depends on the trend running following time and some seasonal effects showing by the factors. And that means it can forecast more than one period ahead. Considering Autocorrelation model, it depends on the lag and lag1 data. It means it can forecast third periods ahead because from its equation has to forecast following the lag data.

4. Try a Box-Jenkins ARIMA approach on the same data and compare your forecasts with the rest of the data as before.

The trendcycle data from the question two will be used to analyse on the Box-Jenkins models as the seasonal effect was removed and the trendcycle data is shown at the chart below.

See the ACF and PACF of the trend-cycle data

From ACF, it shows a trend, so we will do the first difference to remove the trend

From Partial ACF, there is one strong spike on lag one and it dies away after one spike so if a model is fitted to the data, it should be an AR(1).

After first different, ACF does not die away so try the Partial ACF.

This PACF does not show any evidence of pattern repeat and does not die away so try the second difference.

The ACF and PACF of the second difference do not give anything helpful.

Therefore, AR(1) model is suggested by the Partial ACF of the trendcycle data. So we will try ARIMA(1,0,0) trendcycle data.

Model Description

Variable TRENCYC

Regressors NONE

Non-seasonal differencing 0

No seasonal component in model.

Parameters

AR1 ________ value originating from estimation

CONSTANT ________ value originating from estimation

5.00 percent confidence intervals will be generated.

Split group number 1 Series length 6

No missing data.

Melards algorithm will be used for estimation.

Termination criteria

Parameter epsilon .001

Maximum Marquardt constant 1.00E+0

SSQ Percentage .001

Maximum number of iterations 10

Initial values

AR1 .416

CONSTANT 808.40

Conclusion of estimation phase.

Estimation terminated at iteration number because

Sum of squares decreased by less than .001 percent.

FINAL PARAMETERS

Number of residuals 6

Standard error 5.65186

Log likelihood -74.7657

AIC 150.551

SBC 1508.658

Variables in the Model

B SEB T-RATIO APPROX. PROB.

AR1 .8057 .0185 4.8788 .00000000

CONSTANT 8144.71650 17.450 .7476 .00008

The constant term is significant so we can fit an AR(1) to the data with the constant.

The following new variables are being created

Name Label

FIT_1 Fit for TRENCYC from ARIMA, MOD_1 CON

ERR_1 Error for TRENCYC from ARIMA, MOD_1 CON

LCL_1 5% LCL for TRENCYC from ARIMA, MOD_1 CON

UCL_1 5% UCL for TRENCYC from ARIMA, MOD_1 CON

SEP_1 SE of fit for TRENCYC from ARIMA, MOD_1 CON

From the box of variables in the model above, T-ratio of the AR1 and constant are far away from 0 and bigger than and also the all probabilities of which are significant.

So, the ARIMA Model will be

Z t = 8144.71650 + 0.8057 Z t-1

Because of using the trendcycle data with ARIMA, the ARIMA forecast data need to be multiplied by factors in order to get the real forecasts and compare to the actual data

Forecast = (8144.71650 + 0.8057 Z t-1) factors

After multiplying the forecast data by factors and plotting the forecast data against the actual data, it can be seen that it is very close to the data all over the period of time so this ARIMA model fits to the actual data.

Then check this model by considering the residuals by plotting the error. So, the graph shows removing trend and it is stationary.

So, the graph shows removing trend and it is relatively stationary. Then check with ACF and PACF which do not suggest anything.

All figures of ARIMA forecasting are shown as following table.

Time Data Forecast Error LCL UCL

14 Jan 1 41 777.48 -78.40 755.1 1554.1

Feb 80 40.1 -46.50 40.41 55.8

Mar 487 4446.0 4.10 67.1 566.61

Apr 4 46 484.61 -456.74 76.75 61.16

May 5 4657 460.8 54.08 77.10 576.5

Jun 6 457 477.15 180.5 4.85 585.6

Jul 7 464 510.58 -410.17 74. 610.4

Aug 8 551 481. 666.46 404.1 576.7

Sep 48 506.71 -17.71 417.01 6486.4

Oct 10 4688 5050. -6. 870.61 60.0

Nov 11 505 4755.17 6.8 575.46 54.88

Dec 1 57 516.58 5. . 61.80

15 Jan 1 481 46.11 -156.6 404.57 640.8

Feb 14 4571 4664.78 -10.06 46.4 605.84

Mar 15 5485 508.7 401.7 04.0 66.44

Apr 16 508 545.5 -4.64 456.7 6716.

May 17 56 511.81 76.4 407.51 648.

Jun 18 5678 578.0 -117.7 4504.6 6864.11

Jul 1 5587 5841.4 -44.65 447.05 676.47

Aug 0 6116 567. 8.77 446.8 6605.70

Sep 1 556 586.84 -07.84 460.1 704.55

Oct 576 561.18 150.8 44.48 671.8

Nov 5781 580.8 -8.8 46.57 688.

Dec 4 5855 6060.01 -17.1 4647. 7006.64

16 Jan 5 576 54.74 86.5 448.7 6858.8

Feb 6 5448 5556.16 -118.86 45.6 785.8

Mar 7 577 608.74 -55.74 484.04 708.45

Apr 8 650 570.70 660.51 46.8 68.7

May 658 6577.07 4.88 5.4 761.65

Jun 0 60 667.4 -6.76 568.76 778.17

Jul 1 7066 616.6 86.1 4778. 717.75

Aug 6841 7161.50 -05. 5640.76 8000.18

Sep 6480 6546.0 -66.0 567.1 776.61

Oct 4 617 651.5 404.65 5.64 76.06

Nov 5 6468 640.86 -47.86 5761.15 810.56

Dec 6 7161 6760.60 85.00 50.87 7680.

17 Jan 7 6 6564.54 -181.6 570.4 808.75

Feb 8 6088 6147.1 -65.8 5576. 75.65

Mar 646 6718.7 -47.7 558.67 788.08

Apr 40 78 6157.4 14.76 510.1 746.60

May 41 700 761. -604.4 664.84 874.5

Jun 4 747 710.8 6.45 578.1 814.7

Jul 4 7 766.06 75.0 616.66 85.08

Aug 44 7516 800. -470.40 6448.7 8808.1

Sep 45 750 7177.7 4.7 57.56 856.7

Oct 46 74 75.14 410.86 65.4 8711.85

Nov 47 71 746. -555. 6767.1 16.6

Dec 48 8751 7701.87 1008.78 65.4 8585.5

18 Jan 4 7516 788.7 -47.60 7.47 588.8

Feb 50 77 70.66 146.5 676.40 05.8

Mar 51 86 8064. 8.77 6884.5 4.4

Apr 5 8180 811.58 -11.8 717.05 58.47

May 5 87 846.44 -17.46 716.0 5.7

Jun 54 884 808.44 66.1 665.8 5.4

Jul 55 17 146.76 -1.00 7615.5 74.67

Aug 56 8606 01.88 -567.51 758. 4.40

Sep 57 4 815.1 1046.81 7015.48 74.0

Oct 58 878 0.68 -4.68 8040.7 10400.

Nov 5 80 8716.67 0. 756.6 86.7

Dec 60 445 61.1 176.80 775. 10084.64

1 Jan 61 8107 8610.5 -5.84 788.8 104.

Feb 6 84 7758.77 511.4 746.4 705.8

Mar 6 767 00.0 746.8 7840.1 101.7

Apr 64 88 540.77 -7.1 8555.77 1015.1

May 65 8511 086.5 -56.66 7816.68 10176.10

Jun 66 874 858.68 15.14 741.54 600.6

Jul 67 85 1006.57 -177.48 8470.84 1080.6

Aug 68 80 1.6 -85.6 867.56 1066.7

Sep 6 87 8.5 648.75 8158.55 10517.6

Oct 70 460 51.0 -41.0 8771.4 1110.1

Nov 71 1057 44.44 116.56 854.7 10614.15

Dec 7 10450 1071. -501.7 6.64 117.06

000 Jan 7 50 510.54 0.4 881.8 1110.80

Feb 74 1006 05.7 10.67 8815.1 11174.6

Mar 75 108 1061.76 -18.76 78.05 1141.47

Apr 76 4 10565.15 -147.0 601.06 1160.47

May 77 11156 601.75 158.86 86.8 10686.

Jun 78 10806 1108.6 -5.06 80.47 1168.8

Jul 7 1077 1068.8 -187.87 66.8 1176.

Aug 80 1088 1081.4 48.17 15. 1145.4

Sep 81 06 100.6 -414.6 140.8 11500.40

Oct 8 10857 871.78 85. 86.07 11051.48

Nov 8 108 10804.0 87.70 64.5 1184.01

Dec 84 1070 117.16 -541.50 658.1 1018.

001 Jan 85 1114 74.5 1474.18 075.58 1145.00

Feb 86 100 10610.4 -645.5 10480.11 18.5

Mar 87 1106 1058.5 10.47 778.8 118.4

Apr 88 117 10785.1 450.81 85.61 1185.0

May 8 1188 11505.70 88.4 101.07 1571.4

Jun 0 1171 114.75 -14.75 105.85 188.7

Jul 1 1 115.18 8. 101.74 167.15

Aug 15 165.85 -610. 11168.7 158.14

Sep 1158 11668.7 -10.7 10488.56 1847.8

Oct 4 147 115.56 151.44 10115.85 1475.7

Nov 5 184 1147.86 -8.86 1168.15 147.57

Dec 6 165 167.8 -608.55 11577.88 17.0

Comparing the forecast data with the rest of the data, we calculate them by extending the time line and see the result as following table

Time Data Forecast Error LCL UCL

00 Jan 7 176 1664.74 10.8 117.4046 14.04

Feb 8 1165 10.5 -147.8 11764.648 1488.006

Mar 167 14.08 -715.08 1186.16 145.887

Apr 100 184 184.01 714.8 10.184 15087.04

May 101 14085 11.54 6.10 11477.54 14541.5

Jun 10 111 18.7 -146.05 11588.6446 1465.4704

Jul 10 147 1484.8 5.60 1.05 156.046

Aug 104 115 1700.5 -46.8 11516.186 14580.01

Sep 105 18 1587.1 -58.1 1055.04 1511.007

Oct 106 144 1448.75 45.5 1116.868 1480.666

Nov 107 14 175.48 -.48 1174.5704 14807.6

Dec 108 1 1400.47 -78.4 11.66 146.7884

00 Jan 10 158 1646.74 1001. 11780.4406 14844.664

Feb 110 17 1184.77 41.47 1146.5 1456.161

Sum of squared errors = 55.1

Mean-squared error (MSE) = 4566.7

Root-mean-squared error (RMSE) = 68.15

This RMSE is about 4.76 % of the mean for the rest of data during the forecast period

However, overall the plot graph below (During 14-00) shows that the predict data relatively fits the real data.

5. Discuss the differences between the Box-Jenkins results and the other methods you have used. In particular comment on the different mathematical forms of the models selected.

Decomposition Model

Forecast = Trend Factors

= (45.164 + 84.57 Time) Factors

Autocorrelation Model

Forecast = 0.4 (Data lagged) + 0.651 (Data lagged1)

ARIMA Model

Forecast (Zt) = (8144.71650 + 0.8057 Z t-1) Factors

First of all, decomposition model is a linear model which depends on the trendcycle, runs following time, and seasonal effects showing by the factors. So it can forecast more than one period ahead.

Second, autocorrelation model is based on two explanatory variables which are data lagged and data lagged1. So it can forecast third periods ahead because from its equation has to forecast following the lag data.

Finally, Box-Jenkins model or ARIMA model shows the form of an ARIMA(1,0,0) or AR(1) model which depends on the constant term and Zt-1 as well as factors.

The graph below illustrates the comparisons of actual data and three forecasting data which are generated from decomposition model, autocorrelation model and ARIMA model.

It is very difficult to say which model is better because most forecast data are relatively close to real data and some periods one is fitter to the actual data more than others but some periods do contrast. Therefore, we can not tell much different among them.

Next, we will focus on the most appropriate of these three models by considering the error.

Compare the errors of the data for model fit.

From the graph above, it can be seen that the autocorrelation model is more appropriate than other two because of more accuracy. Moreover, it involves only the lag data and does not have any influences from the factors like other two models. If you look at the actual data graph below, it shows a trend pattern but does not show a seasonal pattern. It is emphasised to use autocorrelation model. On the other hand, if the graph shows obviously seasonal pattern, decomposition model should be better. And because of using the trendcycle data to use for ARIMA forecasting, it involves seasonal effects (factors); so make ARIMA model not a proper method as well.

However, there has some lost data at the beginning and at the end by using autocorrelation model to forecast because its data is need to be lagged and it can not forecast a bit more future like Box-Jenkins or ARIMA model.

6. Comment on the impact of your choice of where to split the data in order to use some data to fit the model and other data to check it.

The data at the end (00-00) has been used to check the forecast. From the graph below, we can see that at the beginning and middle of the rest data, the forecast data from autocorrelation model and ARIMA model is close to data but other periods, significant errors will occur. However, there are considerable errors occurred.

Then, trying new data model, we choose data model from 15-00 and data in 1-14 is used for checking. The graph below shows errors from old data (split at the end) against errors from new data (split at the beginning)

The magnitudes of errors from two ways to split data all over the period are similar but they do not overlap at the same time so it does some impact on where to split the data. However, it should be consider the pattern of actual data and choose the appropriate method to forecast rather than think about where to split the data.

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